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Fix X_dot formatting
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briandesilva committed Apr 10, 2020
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14 changes: 13 additions & 1 deletion examples/2_introduction_to_sindy.ipynb
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Expand Up @@ -44,7 +44,19 @@
"## Approximation problem\n",
"To apply SINDy in practice one needs a set of measurement data collected at times $t_1, t_2, \\dots, t_n$, and the time derivatives of these measurements (either measured directly or numerically approximated). These data are aggregated into the matrices $X$ and $\\dot X$, respectively\n",
"\n",
"$$ X = \\begin{bmatrix} x_1(t_1) & x_2(t_1) & \\dots & x_n(t_1) \\\\ x_1(t_2) & x_2(t_2) & \\dots & x_n(t_2) \\\\ \\vdots & \\vdots & & \\vdots \\\\ x_1(t_m) & x_2(t_m) & \\dots & x_n(t_m) \\\\ \\end{bmatrix}, \\qquad \\dot X = \\begin{bmatrix} \\dot{x_1}(t_1) & \\dot{x_2}(t_1) & \\dots & \\dot{x_n}(t_1) \\\\ \\dot{x_1}(t_2) & \\dot{x_2}(t_2) & \\dots & \\dot{x_n}(t_2) \\\\ \\vdots & \\vdots & & \\vdots \\\\ \\dot{x_1}(t_m) & \\dot{x_2}(t_m) & \\dots & \\dot{x_n}(t_m) \\\\ \\end{bmatrix}. $$\n",
"$$\n",
"\tX = \\begin{bmatrix}\n",
"\t\tx_1(t_1) & x_2(t_1) & \\dots & x_n(t_1) \\\\\n",
"\t\tx_1(t_2) & x_2(t_2) & \\dots & x_n(t_2) \\\\\n",
"\t\t\\vdots & \\vdots & & \\vdots \\\\ x_1(t_m) & x_2(t_m) & \\dots & x_n(t_m)\n",
"\t\\end{bmatrix},\n",
"\t\\qquad\n",
"\t\\dot{X} = \\begin{bmatrix} \\dot{x_1}(t_1) & \\dot{x_2}(t_1) & \\dots & \\dot{x_n}(t_1) \\\\\n",
"\t\t\\dot{x_1}(t_2) & \\dot{x_2}(t_2) & \\dots & \\dot{x_n}(t_2) \\\\\n",
"\t\t\\vdots & \\vdots & & \\vdots \\\\\n",
"\t\t\\dot{x_1}(t_m) & \\dot{x_2}(t_m) & \\dots & \\dot{x_n}(t_m)\n",
"\t\\end{bmatrix}.\n",
"$$\n",
"\n",
"Next, one forms a library matrix $\\Theta(X)$ whose columns consist of a chosen set of basis functions applied to the data\n",
"\n",
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